\documentclass[../main.tex]{subfiles}
\begin{document}
% \section{Triangles}
\section{定义与绘制三角形}

% \subsection{Definition of triangles \tkzcname{tkzDefTriangle}}
\subsection{\tkzcname{tkzDefTriangle}命令：定义三角形}
% The following macros will allow you to define or construct a triangle from
% \tkzname{at least} two points.
三角形定义命令允许使用\tkzname{至少}2个点构造一个三角形。
可以按如下方式定义三角形：
% \begin{itemize}
% \item \tkzname{two angles} determines a triangle with two angles;
%
% \item \tkzname{equilateral} determines an equilateral triangle;
%
% \item \tkzname{half} determines a right-angled triangle such that the ratio of
% the measurements of the two adjacent sides to the right angle is equal to $2$;
%
% \item \tkzname{pythagore} determines a right-angled triangle whose side
% measurements are proportional to $3$, $4$ and $5$;
%
% \item \tkzname{school} determines a right-angled triangle whose angles are $30$,
% $60$ and $90$ degrees;
%
% \item \tkzname{golden} determines a right-angled triangle such that the ratio of
% the measurements on the two adjacent sides to the right angle is equal to
% $\Phi=1.618034$, I chose \enquote{golden triangle} as the denomination because it comes
% from the golden rectangle and I kept the denomination \enquote{golden triangle} or
% \enquote{Euclid's triangle} for the isosceles triangle whose angles at the base are $72$
% degrees;
%
% \item \tkzname{euclide} or \tkzname{gold} for the gold triangle;
%
% \item \tkzname{cheops} determines a third point such that the triangle is
% isosceles with side measurements proportional to $2$, $\Phi$ and $\Phi$.
% \end{itemize}

\begin{itemize}
\item \tkzname{two angles}已知2个角的三角形;

\item \tkzname{equilateral}等边三角形;

\item \tkzname{half}直角边之和与斜边之比等于$2$的直角三角形;

\item \tkzname{pythagore} 勾股直角三角形;

\item \tkzname{school} 三个角分别是$30$、$60$和$90$的直角三角形;

\item \tkzname{golden}直角边之和与斜边比等于黄金分割比$\Phi=1.618034$的直角三角形，
;

\item \tkzname{euclide} or \tkzname{gold}
底角为$72$度的等腰三角形称为\enquote{golden triangle}或\enquote{Euclid's triangle};

\item \tkzname{cheops} 三边为$2$、$\Phi$和 $\Phi$的等腰三角形.
\end{itemize}

% \begin{NewMacroBox}{tkzDefTriangle}{\oarg{local options}\parg{A,B}}%
%
% The points are ordered because the triangle is constructed following the direct
% direction of the trigonometric circle. This macro is either used in partnership
% with \tkzcname{tkzGetPoint} or by using \tkzname{tkzPointResult} if it is not
% necessary to keep the name.
%
% \medskip
% \begin{tabular}{lll}%
% \toprule
% options             & default & definition                        \\
% \midrule
% \TOline{two angles= \#1 and \#2}{no defaut}{triangle knowing two angles}
% \TOline{equilateral} {no defaut}{equilateral triangle }
% \TOline{pythagore}{no defaut}{proportional to the pythagorean triangle $3-4-5$}
% \TOline{school} {no defaut}{angles of $30$, $60$ and $90$ degrees }
% \TOline{gold}{no defaut}{angles of $72$, $72$ and $36$ degrees, $A$ is the apex}
% \TOline{euclide} {no defaut}{same as above but $[AB]$ is the base}
% \TOline{golden} {no defaut}{$B$ rectangle and $AB/AC = \Phi$}
% \TOline{cheops} {no defaut}{$AC=BC$, $AC$ and $BC$ are proportional to $2$ and $\Phi$.}
% \bottomrule
% \end{tabular}
%
% \medskip
% \tkzcname{tkzGetPoint} allows you to store the point otherwise
% \tkzname{tkzPointResult} allows for immediate use.
%
% \end{NewMacroBox}
\begin{NewMacroBox}{tkzDefTriangle}{\oarg{命令选项}\parg{A,B}}%
参数中的点的顺序决定的另外一个点的位置。
可使用\tkzcname{tkzGetPoint}保存并命令定义的点，
也可使用\tkzname{tkzPointResult}命令使用计算结果，但不命名。

\medskip
\begin{tabular}{lll}%
\toprule
选项             & 默认值 & 含义                        \\
\midrule
\TOline{two angles= \#1 and \#2}{无}{三角形两个已知角}
\TOline{equilateral} {无}{等边三角形}
\TOline{pythagore}{无}{勾股三角形}
\TOline{school} {无}{三个角分别是$30$、$60$和$90$度}
\TOline{gold}{无}{三个角分别是$72$、$72$和$36$度，$A$是顶点}
\TOline{euclide} {无}{同上，但$[AB]$是底边}
\TOline{golden} {无}{$AB$构成矩形，并且$AB/BC = \Phi$}
\TOline{cheops} {无}{$AC=BC$，$AC$和$BC$及第三边满足$2$和$\Phi$的比例关系}
\bottomrule
\end{tabular}

\medskip
使用\tkzcname{tkzGetPoint}保存并命名得到的点，
使用\tkzname{tkzPointResult}命令使用得到的点，但不命名。

\end{NewMacroBox}

\vspace*{-10pt}

% \subsubsection{Option \tkzname{golden}}
\subsubsection{\tkzname{golden}选项示例}

\begin{tkzexample}[latex=6cm,small]
\begin{tikzpicture}[scale=0.9]
  \tkzInit[xmax=5,ymax=3]
  \tkzClip[space=.5]
  \tkzDefPoint(0,0){A} \tkzDefPoint(4,0){B}
  \tkzDefTriangle[golden](A,B)\tkzGetPoint{C}
  \tkzDrawPolygon(A,B,C) \tkzDrawPoints(A,B,C)
  \tkzLabelPoints(A,B) \tkzDrawBisector(A,C,B)
  \tkzLabelPoints[above](C)
\end{tikzpicture}
\end{tkzexample}

\newpage

% \subsubsection{Option \tkzname{equilateral}}
\subsubsection{\tkzname{equilateral}选项示例}

\begin{tkzexample}[latex=7cm,small]
\begin{tikzpicture}
  \tkzDefPoint(0,0){A}
  \tkzDefPoint(3,0){B}
  \tkzDefTriangle[equilateral](A,B)
  \tkzGetPoint{C}
  \tkzDrawPolygon(A,B,C)
  \tkzDefTriangle[equilateral](B,A)
  \tkzGetPoint{D}
  \tkzDrawPolygon(B,A,D)
  \tkzDrawPoints(A,B,C,D)
  \tkzLabelPoints(A,B,C,D)
\end{tikzpicture}
\end{tkzexample}

% \subsubsection{Option \tkzname{gold} or \tkzname{euclide}}
\subsubsection{\tkzname{gold}或\tkzname{euclide}选项示例}

\begin{tkzexample}[latex=7cm,small]
\begin{tikzpicture}[scale=0.75]
  \tkzDefPoint(0,0){A}
  \tkzDefPoint(4,0){B}
  \tkzDefTriangle[euclide](A,B)
  \tkzGetPoint{C}
  \tkzDrawPolygon(A,B,C)
  \tkzDrawPoints(A,B,C)
  \tkzLabelPoints(A,B)
  \tkzLabelPoints[above](C)
  \tkzDrawBisector(A,C,B)
\end{tikzpicture}
\end{tkzexample}

% \subsection{Drawing of triangles}
\subsection{\tkzcname{tkzDrawTriangle}命令：绘制三角形}

% \begin{NewMacroBox}{tkzDrawTriangle}{\oarg{local options}\parg{A,B}}%
% Macro similar to the previous macro but the sides are drawn.
%
% \medskip
% \begin{tabular}{lll}%
% \toprule
% options             & default & definition                        \\
% \midrule
% \TOline{two angles= \#1 and \#2}{equilateral}{triangle knowing two angles}
% \TOline{equilateral} {equilateral}{equilateral triangle }
% \TOline{pythagore}{equilateral}{proportional to the pythagorean triangle 3-4-5}
% \TOline{school} {equilateral}{the angles are 30, 60 and 90 degrees }
% \TOline{gold}{equilateral}{the angles are 72, 72 and 36 degrees, $A$ is the vertex }
% \TOline{euclide} {equilateral}{identical to the previous one but $[AB]$ is the base}
% \TOline{golden} {equilateral}{B rectangle and $AB/AC = \Phi$}
% \TOline{cheops} {equilateral}{isosceles in C and $AC/AB = \frac{\Phi}{2}$}
% \bottomrule
% \end{tabular}
%
% \medskip
% In all its definitions, the dimensions of the triangle depend on the two
% starting points.
% \end{NewMacroBox}
\begin{NewMacroBox}{tkzDrawTriangle}{\oarg{命令选项}\parg{A,B}}%
与三角形定义命令类似，但可绘制三角形。

\medskip
\begin{tabular}{lll}%
\toprule
选项             & 默认值 & 含义                        \\
\midrule
\TOline{two angles= \#1 and \#2}{无}{三角形两个已知角度}
\TOline{equilateral} {无}{等边三角形}
\TOline{pythagore}{无}{勾股三角形}
\TOline{school} {无}{三个角分别是$30$、$60$和$90$度}
\TOline{gold}{无}{三个角分别是$72$、$72$和$36$度，$A$是顶点}
\TOline{euclide} {无}{同上，但$[AB]$是底边}
\TOline{golden} {无}{$AB$构成矩形，并且$AB/AC = \Phi$}
\TOline{cheops} {无}{$AC=BC$，$AC$和$BC$及第三边满足$2$和$\Phi$的比例关系}
\bottomrule
\end{tabular}

\medskip
定义中，三角形尺寸取决于起始的两个点。
\end{NewMacroBox}

\newpage

% \subsubsection{Option \tkzname{pythagore}}
\subsubsection{\tkzname{pythagore}选项示例}

% This triangle has sides whose lengths are proportional to $3$, $4$ and $5$.
三角形的三个边满足类似$3$、$4$和$5$的勾股关系。

\begin{tkzexample}[latex=6cm,small]
\begin{tikzpicture}
  \tkzDefPoint(0,0){A}
  \tkzDefPoint(4,0){B}
  \tkzDrawTriangle[pythagore,fill=blue!30](A,B)
  \tkzMarkRightAngles(A,B,tkzPointResult)
\end{tikzpicture}
\end{tkzexample}

% \subsubsection{Option \tkzname{school}}
\subsubsection{\tkzname{school}选项示例}

% The angles are $30$, $60$ and $90$ degrees.
三角形的三个内角分别是$30$、$60$和$90$度。

\begin{tkzexample}[latex=6cm,small]
\begin{tikzpicture}
  \tkzDefPoint(0,0){A}
  \tkzDefPoint(4,0){B}
  \tkzDrawTriangle[school,fill=red!30](A,B)
  \tkzMarkRightAngles(tkzPointResult,B,A)
\end{tikzpicture}
\end{tkzexample}

% \subsubsection{Option \tkzname{golden}}
\subsubsection{\tkzname{golden}选项示例}

\begin{tkzexample}[latex=6cm,small]
\begin{tikzpicture}[scale=1]
  \tkzDefPoint(0,-10){M}
  \tkzDefPoint(3,-10){N}
  \tkzDrawTriangle[golden,color=brown](M,N)
\end{tikzpicture}
\end{tkzexample}

% \subsubsection{Option \tkzname{gold}}
\subsubsection{\tkzname{gold}选项示例}

\begin{tkzexample}[latex=6cm,small]
\begin{tikzpicture}[scale=1]
  \tkzDefPoint(5,-5){I}
  \tkzDefPoint(8,-5){J}
  \tkzDrawTriangle[gold,color=blue!50](I,J)
\end{tikzpicture}
\end{tkzexample}

% \subsubsection{Option \tkzname{euclide}}
\subsubsection{\tkzname{euclide}选项示例}

\begin{tkzexample}[latex=6cm,small]
\begin{tikzpicture}[scale=1]
  \tkzDefPoint(10,-5){K}
  \tkzDefPoint(13,-5){L}
  \tkzDrawTriangle[euclide,color=blue,fill=blue!10](K,L)
\end{tikzpicture}
\end{tkzexample}

\newpage

% \section{Specific triangles with \tkzcname{tkzDefSpcTriangle}}
\subsection{\tkzcname{tkzDefSpcTriangle}命令：定义特殊三角形}

% The centers of some triangles have been defined in the \enquote{points} section,
% here it is a question of determining the three vertices of specific triangles.
在\enquote{点}的定义小节中，定义了一些三角形中的特殊点，
在此，可以使用这些点确定三角形。

% \begin{NewMacroBox}{tkzDefSpcTriangle}{\oarg{local options}\parg{A,B,C}}
% The order of the points is important!
%
% \medskip
% \begin{tabular}{lll}%
% \toprule
% options             & default & definition                        \\
% \midrule
% \TOline{in or incentral}{centroid}{two-angled triangle}
% \TOline{ex or excentral} {centroid}{equilateral triangle }
% \TOline{extouch}{centroid}{proportional to the pythagorean triangle 3-4-5}
% \TOline{intouch or contact} {centroid}{ 30, 60 and 90 degree angles }
% \TOline{centroid or medial}{centroid}{ angles of 72, 72 and 36 degrees, $A$ is the vertex }
% \TOline{orthic} {centroid}{same as above but $[AB]$ is the base}
% \TOline{feuerbach} {centroid}{B rectangle and $AB/AC = \Phi$}
% \TOline{euler} {centroid}{AC=BC, AC and BC are proportional to $2$ and $\Phi$.}
% \TOline{tangential} {centroid}{AC=BC, AC and BC are proportional to $2$ and $\Phi$.}
% \TOline{name} {no defaut}{AC=BC, AC and BC are proportional to $2$ and $\Phi$.}
% \midrule
% \end{tabular}
%
% \medskip
% \tkzcname{tkzGetPoint} allows you to store the point otherwise
% \tkzname{tkzPointResult} allows for immediate use.
% \end{NewMacroBox}
\begin{NewMacroBox}{tkzDefSpcTriangle}{\oarg{命令选项}\parg{A,B,C}}
注意，点的顺序决定了计算结果。

\medskip
\begin{tabular}{lll}%
\toprule
选项             & 默认值 & 含义                        \\
\midrule
\TOline{in or incentral}{centroid}{内心三角形}
\TOline{ex or excentral} {centroid}{旁心三角形 }
\TOline{extouch}{centroid}{外切三角形}
\TOline{intouch or contact} {centroid}{内切三角形}
\TOline{centroid or medial}{centroid}{三边中点构成的三角形}
\TOline{orthic} {centroid}{正交三角形}
\TOline{feuerbach} {centroid}{九点圆与旁切圆的切点构成的三角形}
\TOline{euler} {centroid}{欧拉三角形}
\TOline{tangential} {centroid}{切身三角形}
\TOline{name} {无}{顶点命名前缀}
\midrule
\end{tabular}

\medskip
使用\tkzcname{tkzGetPoint}保存并命名点，
使用\tkzcname{tkzPointResult}命令使用得到的点，但不命名。
\end{NewMacroBox}

% \subsubsection{Option \tkzname{medial} or \tkzname{centroid}}
\subsubsection{\tkzname{medial}或\tkzname{centroid}选项示例}

% The geometric centroid  of the polygon vertices of a triangle is the point $G$
% (sometimes also denoted $M$) which is also the intersection of the triangle's
% three triangle medians. The point is therefore sometimes called the median
% point. The centroid is always in the interior of the triangle.\par
三角形的质心用$G$表示(有时也用$M$表示)，它是三角形三条中线的将点，该点也称为重心，
重心总是位于三角形内部。\par
\href{https://mathworld.wolfram.com/TriangleCentroid.html}{Weisstein, Eric W.
\enquote{Centroid triangle} From MathWorld--A Wolfram Web Resource.}

% In the following example, we obtain the Euler circle which passes through the
% previously defined points.
下面的例子中，通过预先定义的点，得到通过这些点的欧拉圆。

\begin{tkzexample}[latex=7cm,small]
\begin{tikzpicture}[rotate=90,scale=1.15]
  \tkzDefPoints{0/0/A,6/0/B,0.8/4/C}
  \tkzDefTriangleCenter[centroid](A,B,C)
  \tkzGetPoint{M}
  \tkzDefSpcTriangle[medial,name=M](A,B,C){_A,_B,_C}
  \tkzDrawPolygon[color=blue](A,B,C)
  \tkzDrawSegments[dashed,red](A,M_A B,M_B C,M_C)
  \tkzDrawPolygon[color=red](M_A,M_B,M_C)
  \tkzDrawPoints(A,B,C,M)
  \tkzDrawPoints[red](M_A,M_B,M_C)
  \tkzAutoLabelPoints[center=M,font=\scriptsize]%
(A,B,C,M_A,M_B,M_C)
  \tkzLabelPoints[font=\scriptsize](M)
\end{tikzpicture}
\end{tkzexample}

\newpage

% \subsubsection{Option \tkzname{in} or \tkzname{incentral}}
\subsubsection{\tkzname{in}或\tkzname{incentral}选项示例}

% The incentral triangle is the triangle whose vertices are determined by
% the intersections of the reference triangle’s angle bisectors with the
% respective opposite sides.\par
内心三角形是由一个三角形的三个内角平分线与对边交点确定的三角形。\par
\href{https://mathworld.wolfram.com/ContactTriangle.html}{Weisstein, Eric W.
\enquote{Incentral triangle} From MathWorld--A Wolfram Web Resource.}

\begin{tkzexample}[latex=7cm,small]
\begin{tikzpicture}[scale=1.15]
  \tkzDefPoints{ 0/0/A,5/0/B,1/3/C}
  \tkzDefSpcTriangle[in,name=I](A,B,C){_a,_b,_c}
  \tkzInCenter(A,B,C)
  \tkzGetPoint{I}
  \tkzDrawPolygon[red](A,B,C)
  \tkzDrawPolygon[blue](I_a,I_b,I_c)
  \tkzDrawPoints(A,B,C,I,I_a,I_b,I_c)
  \tkzDrawCircle[in](A,B,C)
  \tkzDrawSegments[dashed](A,I_a B,I_b C,I_c)
  \tkzAutoLabelPoints[center=I,
    blue,font=\scriptsize](I_a,I_b,I_c)
  \tkzAutoLabelPoints[center=I,red,
    font=\scriptsize](A,B,C,I_a,I_b,I_c)
\end{tikzpicture}
\end{tkzexample}

% \subsubsection{Option \tkzname{ex} or \tkzname{excentral} }
\subsubsection{\tkzname{ex}或\tkzname{excentral}选项示例}

% The excentral triangle of a triangle $ABC$ is the triangle $J_aJ_bJ_c$ with
% vertices corresponding to the excenters of $ABC$.
旁心三角形是由一个三角形的三个旁心构成的三角形。

\begin{tkzexample}[latex=7cm,small]
\begin{tikzpicture}[scale=0.60]
  \tkzDefPoints{0/0/A,6/0/B,0.8/4/C}
  \tkzDefSpcTriangle[excentral,name=J](A,B,C){_a,_b,_c}
  \tkzDefSpcTriangle[extouch,name=T](A,B,C){_a,_b,_c}
  \tkzDrawPolygon[blue](A,B,C)
  \tkzDrawPolygon[red](J_a,J_b,J_c)
  \tkzDrawPoints(A,B,C)
  \tkzDrawPoints[red](J_a,J_b,J_c)
  \tkzLabelPoints(A,B,C)
  \tkzLabelPoints[red](J_b,J_c)
  \tkzLabelPoints[red,above](J_a)
  \tkzClipBB
  \tkzShowBB
  \tkzDrawCircles[gray](J_a,T_a J_b,T_b J_c,T_c)
\end{tikzpicture}
\end{tkzexample}

\newpage

% \subsubsection{Option \tkzname{intouch}}
\subsubsection{\tkzname{intouch}选项示例}

% The contact triangle of a triangle $ABC$, also called the intouch triangle, is
% the triangle  formed by the points of tangency of the incircle of $ABC$ with $ABC$.\par
% \href{https://mathworld.wolfram.com/ContactTriangle.html}{Weisstein, Eric W.
% \enquote{Contact triangle} From MathWorld--A Wolfram Web Resource.}
内接三角形是由三角形的内切圆的三个切点构成的三角形。\par
\href{https://mathworld.wolfram.com/ContactTriangle.html}{Weisstein, Eric W.
\enquote{Contact triangle} From MathWorld--A Wolfram Web Resource.}

% We obtain the intersections of the bisectors with the sides.

\begin{tkzexample}[latex=7cm,small]
\begin{tikzpicture}[scale=.75]
  \tkzDefPoints{0/0/A,6/0/B,0.8/4/C}
  \tkzDefSpcTriangle[intouch,name=X](A,B,C){_a,_b,_c}
  \tkzInCenter(A,B,C)\tkzGetPoint{I}
  \tkzDrawPolygon[red](A,B,C)
  \tkzDrawPolygon[blue](X_a,X_b,X_c)
  \tkzDrawPoints[red](A,B,C)
  \tkzDrawPoints[blue](X_a,X_b,X_c)
  \tkzDrawCircle[in](A,B,C)
  \tkzAutoLabelPoints[center=I,blue,font=\scriptsize]%
(X_a,X_b,X_c)
  \tkzAutoLabelPoints[center=I,red,font=\scriptsize]%
(A,B,C)
\end{tikzpicture}
\end{tkzexample}

% \subsubsection{Option \tkzname{extouch}}
\subsubsection{\tkzname{extouch}选项示例}

% The extouch triangle  $T_aT_bT_c$ is the triangle formed by the points of
% tangency of a triangle $ABC$ with its excircles $J_a$, $J_b$, and $J_c$. The
% points  $T_a$, $T_b$, and $T_c$ can also be constructed as the points which
% bisect the perimeter of $A_1A_2A_3$ starting at $A$, $B$, and $C$.\par
外切三角形$T_aT_bT_c$是由三角形$ABC$的三个旁切圆$J_a$、$J_b$和$J_c$的切点构成的三角形。\par
% $T_a$、$T_b$和$T_c$这三个点也可以由从$A$、$B$和$C$开始$A_1A_2A_3$的周长的中点得到。\par
\href{https://mathworld.wolfram.com/ExtouchTriangle.html}{Weisstein, Eric W.
\enquote{Extouch triangle} From MathWorld--A Wolfram Web Resource.}

% We obtain the points of contact of the exinscribed circles as well as the
% triangle formed by the centres of the exinscribed circles.
可以得到旁切圆的切点和由这三个切点构成的三角形。

\begin{tkzexample}[latex=8cm,small]
\begin{tikzpicture}[scale=.7]
  \tkzDefPoints{0/0/A,6/0/B,0.8/4/C}
  \tkzDefSpcTriangle[excentral,
                     name=J](A,B,C){_a,_b,_c}
  \tkzDefSpcTriangle[extouch,
                     name=T](A,B,C){_a,_b,_c}
  \tkzDefTriangleCenter[nagel](A,B,C)
  \tkzGetPoint{N_a}
  \tkzDefTriangleCenter[centroid](A,B,C)
  \tkzGetPoint{G}
  \tkzDrawPoints[blue](J_a,J_b,J_c)
  \tkzClipBB \tkzShowBB
  \tkzDrawCircles[gray](J_a,T_a J_b,T_b J_c,T_c)
  \tkzDrawLines[add=1 and 1](A,B B,C C,A)
  \tkzDrawSegments[gray](A,T_a B,T_b C,T_c)
  \tkzDrawSegments[gray](J_a,T_a J_b,T_b J_c,T_c)
  \tkzDrawPolygon[blue](A,B,C)
  \tkzDrawPolygon[red](T_a,T_b,T_c)
  \tkzDrawPoints(A,B,C,N_a)
  \tkzLabelPoints(N_a)
  \tkzAutoLabelPoints[center=N_a,blue](A,B,C)
  \tkzAutoLabelPoints[center=G,red,
                      dist=.4](T_a,T_b,T_c)
  \tkzMarkRightAngles[fill=gray!15](J_a,T_a,B
 J_b,T_b,C J_c,T_c,A)
\end{tikzpicture}
\end{tkzexample}

\newpage

% \subsubsection{Option \tkzname{feuerbach}}
\subsubsection{\tkzname{feuerbach}选项示例}

% The Feuerbach triangle is the triangle formed by the three points of tangency of
% the nine-point circle with the excircles.\par
Feuerbach三角形是由九点圆与三个旁切圆的3个切点构成的三角形。\par
\href{https://mathworld.wolfram.com/FeuerbachTriangle.html}{Weisstein, Eric W.
\enquote{Feuerbach triangle} From MathWorld--A Wolfram Web Resource.}

% The points of tangency define the Feuerbach triangle.

\begin{tkzexample}[latex=8cm,small]
\begin{tikzpicture}[scale=1.25]
  \tkzDefPoint(0,0){A}
  \tkzDefPoint(3,0){B}
  \tkzDefPoint(0.5,2.5){C}
  \tkzDefCircle[euler](A,B,C) \tkzGetPoint{N}
  \tkzDefSpcTriangle[feuerbach,
     name=F](A,B,C){_a,_b,_c}
  \tkzDefSpcTriangle[excentral,
     name=J](A,B,C){_a,_b,_c}
  \tkzDefSpcTriangle[extouch,
     name=T](A,B,C){_a,_b,_c}
  \tkzDrawPoints[blue](J_a,J_b,J_c,F_a,F_b,F_c,A,B,C)
  \tkzClipBB
  \tkzShowBB
  \tkzDrawCircle[purple](N,F_a)
  \tkzDrawPolygon(A,B,C)
  \tkzDrawPolygon[blue](F_a,F_b,F_c)
  \tkzDrawCircles[gray](J_a,F_a J_b,F_b J_c,F_c)
  \tkzAutoLabelPoints[center=N,dist=.3,
     font=\scriptsize](A,B,C,F_a,F_b,F_c,J_a,J_b,J_c)
\end{tikzpicture}
\end{tkzexample}

% \subsubsection{Option \tkzname{tangential}}
\subsubsection{\tkzname{tangential}选项示例}

% The tangential triangle is the triangle $T_aT_bT_c$ formed by the lines tangent
% to the circumcircle of a given triangle $ABC$ at its vertices. It is therefore
% antipedal triangle of $ABC$ with respect to the circumcenter $O$.\par
切向三角形是三角形$ABC$外接圆在三个顶点处的切线构成的三角形$T_aT_bT_c$。
因此，它是相对于三角形$ABC$外心的反三角三角形。\par
\href{https://mathworld.wolfram.com/TangentialTriangle.html}{Weisstein, Eric W.
\enquote{Tangential Triangle} From MathWorld--A Wolfram Web Resource.}

\begin{tkzexample}[latex=8cm,small]
\begin{tikzpicture}[scale=.5,rotate=80]
  \tkzDefPoints{0/0/A,6/0/B,1.8/4/C}
  \tkzDefSpcTriangle[tangential,
    name=T](A,B,C){_a,_b,_c}
  \tkzDrawPolygon[red](A,B,C)
  \tkzDrawPolygon[blue](T_a,T_b,T_c)
  \tkzDrawPoints[red](A,B,C)
  \tkzDrawPoints[blue](T_a,T_b,T_c)
  \tkzDefCircle[circum](A,B,C)
  \tkzGetPoint{O}
  \tkzDrawCircle(O,A)
  \tkzLabelPoints[red](A,B,C)
  \tkzLabelPoints[blue](T_a,T_b,T_c)
\end{tikzpicture}
\end{tkzexample}

\newpage

% \subsubsection{Option \tkzname{euler}}
\subsubsection{\tkzname{euler}选项示例}

% The Euler triangle of a triangle $ABC$ is the triangle $E_AE_BE_C$ whose
% vertices are the midpoints of the segments joining the orthocenter $H$ with the
% respective vertices. The vertices of the triangle are known as the Euler points,
% and lie on the nine-point circle.
欧拉三角形是由三角形$ABC$的垂心$H$与三个顶点连线中点构成的三角形$E_AE_BE_C$，
欧拉三角形的顶点是欧拉点，它们位于三角形的九点圆上。

\begin{tkzexample}[latex=7cm,small]
\begin{tikzpicture}[rotate=90,scale=1.25]
  \tkzDefPoints{0/0/A,6/0/B,0.8/4/C}
  \tkzDefSpcTriangle[medial,
                     name=M](A,B,C){_A,_B,_C}
  \tkzDefTriangleCenter[euler](A,B,C)
  \tkzGetPoint{N} % I= N nine points
  \tkzDefTriangleCenter[ortho](A,B,C)
  \tkzGetPoint{H}
  \tkzDefMidPoint(A,H) \tkzGetPoint{E_A}
  \tkzDefMidPoint(C,H) \tkzGetPoint{E_C}
  \tkzDefMidPoint(B,H) \tkzGetPoint{E_B}
  \tkzDefSpcTriangle[ortho,name=H](A,B,C){_A,_B,_C}
  \tkzDrawPolygon[color=blue](A,B,C)
  \tkzDrawCircle(N,E_A)
  \tkzDrawSegments[blue](A,H_A B,H_B C,H_C)
  \tkzDrawPoints(A,B,C,N,H)
  \tkzDrawPoints[red](M_A,M_B,M_C)
  \tkzDrawPoints[blue]( H_A,H_B,H_C)
  \tkzDrawPoints[green](E_A,E_B,E_C)
  \tkzAutoLabelPoints[center=N,font=\scriptsize]%
(A,B,C,M_A,M_B,M_C,H_A,H_B,H_C,E_A,E_B,E_C)
  \tkzLabelPoints[font=\scriptsize](H,N)
  \tkzMarkSegments[mark=s|,size=3pt,
         color=blue,line width=1pt](B,E_B E_B,H)
  \tkzDrawPolygon[color=red](M_A,M_B,M_C)
\end{tikzpicture}
\end{tkzexample}

\end{document}
\endinput
